414 research outputs found
Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions
A greedily routable region (GRR) is a closed subset of , in
which each destination point can be reached from each starting point by
choosing the direction with maximum reduction of the distance to the
destination in each point of the path.
Recently, Tan and Kermarrec proposed a geographic routing protocol for dense
wireless sensor networks based on decomposing the network area into a small
number of interior-disjoint GRRs. They showed that minimum decomposition is
NP-hard for polygons with holes.
We consider minimum GRR decomposition for plane straight-line drawings of
graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing
style which has become a popular research topic in graph drawing. We show that
minimum decomposition is still NP-hard for graphs with cycles, but can be
solved optimally for trees in polynomial time. Additionally, we give a
2-approximation for simple polygons, if a given triangulation has to be
respected.Comment: full version of a paper appearing in ISAAC 201
On the spectral dimension of causal triangulations
We introduce an ensemble of infinite causal triangulations, called the
uniform infinite causal triangulation, and show that it is equivalent to an
ensemble of infinite trees, the uniform infinite planar tree. It is proved that
in both cases the Hausdorff dimension almost surely equals 2. The infinite
causal triangulations are shown to be almost surely recurrent or, equivalently,
their spectral dimension is almost surely less than or equal to 2. We also
establish that for certain reduced versions of the infinite causal
triangulations the spectral dimension equals 2 both for the ensemble average
and almost surely. The triangulation ensemble we consider is equivalent to the
causal dynamical triangulation model of two-dimensional quantum gravity and
therefore our results apply to that model.Comment: 22 pages, 6 figures; typos fixed, one extra figure, references
update
Transversal structures on triangulations: a combinatorial study and straight-line drawings
This article focuses on a combinatorial structure specific to triangulated
plane graphs with quadrangular outer face and no separating triangle, which are
called irreducible triangulations. The structure has been introduced by Xin He
under the name of regular edge-labelling and consists of two bipolar
orientations that are transversal. For this reason, the terminology used here
is that of transversal structures. The main results obtained in the article are
a bijection between irreducible triangulations and ternary trees, and a
straight-line drawing algorithm for irreducible triangulations. For a random
irreducible triangulation with vertices, the grid size of the drawing is
asymptotically with high probability up to an additive
error of \cO(\sqrt{n}). In contrast, the best previously known algorithm for
these triangulations only guarantees a grid size .Comment: 42 pages, the second version is shorter, focusing on the bijection
(with application to counting) and on the graph drawing algorithm. The title
has been slightly change
Planar maps and continued fractions
We present an unexpected connection between two map enumeration problems. The
first one consists in counting planar maps with a boundary of prescribed
length. The second one consists in counting planar maps with two points at a
prescribed distance. We show that, in the general class of maps with controlled
face degrees, the solution for both problems is actually encoded into the same
quantity, respectively via its power series expansion and its continued
fraction expansion. We then use known techniques for tackling the first problem
in order to solve the second. This novel viewpoint provides a constructive
approach for computing the so-called distance-dependent two-point function of
general planar maps. We prove and extend some previously predicted exact
formulas, which we identify in terms of particular Schur functions.Comment: 47 pages, 17 figures, final version (very minor changes since v2
Generic Ising Trees
The Ising model on an infinite generic tree is defined as a thermodynamic
limit of finite systems. A detailed description of the corresponding
distribution of infinite spin configurations is given. As an application we
study the magnetization properties of such systems and prove that they exhibit
no spontaneous magnetization. Furthermore, the values of the Hausdorff and
spectral dimensions of the underlying trees are calculated and found to be,
respectively, and .Comment: 29 pages, 2 figures; typos corrected, one section and new references
adde
Local limit of labeled trees and expected volume growth in a random quadrangulation
Exploiting a bijective correspondence between planar quadrangulations and
well-labeled trees, we define an ensemble of infinite surfaces as a limit of
uniformly distributed ensembles of quadrangulations of fixed finite volume. The
limit random surface can be described in terms of a birth and death process and
a sequence of multitype Galton--Watson trees. As a consequence, we find that
the expected volume of the ball of radius around a marked point in the
limit random surface is .Comment: Published at http://dx.doi.org/10.1214/009117905000000774 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Some families of increasing planar maps
Stack-triangulations appear as natural objects when one wants to define some
increasing families of triangulations by successive additions of faces. We
investigate the asymptotic behavior of rooted stack-triangulations with
faces under two different distributions. We show that the uniform distribution
on this set of maps converges, for a topology of local convergence, to a
distribution on the set of infinite maps. In the other hand, we show that
rescaled by , they converge for the Gromov-Hausdorff topology on
metric spaces to the continuum random tree introduced by Aldous. Under a
distribution induced by a natural random construction, the distance between
random points rescaled by converge to 1 in probability.
We obtain similar asymptotic results for a family of increasing
quadrangulations
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